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In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called interval spheres. In this paper, we provide an algorithm to decompose filtered chain complexes into such interval spheres. This algorithm provides geometric insights into various aspects of the standard persistence algorithm and two of its run-time optimizations. Moreover, since it works for any filtered chain complexes, our algorithm can be applied in more general cases. As an application, we show how to decompose filtered kernels with it., Revised version, decomposition of filtered kernels added

In this work, we provide a model structure on full subcategories of tame objects in functor categories indexed by continuous realizations of posets of dimension 1. We also characterize the indecomposable cofibrant objects when the landing category is the one of bounded non-negative chain complexes. In addition, we present a general method to construct indecomposables in a functor category by a gluing technique., Comment: 15 pages, 4 figures

We set the foundations for a new approach to Topological Data Analysis (TDA) based on homotopical methods at chain complexes level. We present the category of tame parametrised chain complexes as a comprehensive environment that includes several cases that usually TDA handles separately, such as persistence modules, zigzag modules, and commutative ladders. We extract new invariants in this category using a model structure and various minimal cofibrant approximations. Such approximations and their invariants retain some of the topological, and not just homological, aspects of the objects they approximate.

Environmental cues influence the highly dynamic morphology of microglia. Strategies to characterize these changes usually involve user-selected morphometric features, which preclude the identification of a spectrum of context-dependent morphological phenotypes. Here we develop MorphOMICs, a topological data analysis approach, which enables semiautomatic mapping of microglial morphology into an atlas of cue-dependent phenotypes and overcomes feature-selection biases and biological variability. We extract spatially heterogeneous and sexually dimorphic morphological phenotypes for seven adult mouse brain regions. This sex-specific phenotype declines with maturation but increases over the disease trajectories in two neurodegeneration mouse models, with females showing a faster morphological shift in affected brain regions. Remarkably, microglia morphologies reflect an adaptation upon repeated exposure to ketamine anesthesia and do not recover to control morphologies. Finally, we demonstrate that both long primary processes and short terminal processes provide distinct insights to morphological phenotypes. MorphOMICs opens a new perspective to characterize microglial morphology.

In the last decade, there has been an unprecedented development in topological data analysis tools to encode geometrical information. These tools have been successful for extracting new information from brain networks with a range of methods such as persistent homology, which provides a framework for obtaining higher order topological features from the brain that go beyond modular structures, and include homological significant aspects such as number of holes and cycles. In this talk I will give an introduction to what these techniques have to offer and present illustrative examples. I will focus particularly on how to extract statistically viable conclusions based on combining topological data analysis methods with subsampling techniques and discuss potential applications of these methods in the context of neurodegenerative diseases. This abstract is part of the symposium: "Diagnosis and Prediction of Neurodegenerative Diseases using Artificial Intelligence and Simulations".

The relationship between the structure and network dynamics within the striatum is currently not well understood. We have applied algebraic topology to investigate the local structural connectivity in the striatum, and then used simulations to predict how structure shapes network dynamics. We used a full-scale digital reconstruction of the mouse striatal microcircuitry: both healthy and at different stages of Parkinson’s Disease (PD). These stages are characterized by successively modified healthy morphologies of the striatal projection neurons (SPN), including changes in dendritic spine count. We compared the distribution of topological motifs, in the form of directed cliques, between these microcircuits. The distribution of directed cliques in the healthy striatal microcircuits showed that striatal interneurons, despite only accounting for 5%, are crucial for the construction of high dimensional directed cliques. In PD networks the presence of directed cliques drastically decreased with the disease progression. We then used simulations to investigate whether these changes in structural connectivity affect functional connectivity. Signal transfer, especially correlation transfer, in the corticostriatal system was affected. We also found that the resulting changes in intrastriatal inhibition altered the correlations between the striatal projection neurons. Directed cliques already provided insight on structural and functional properties of neocortical micrucircuitry. Here we applied this topological approach to investigate striatal networks and highligthed important differences with respect to neocortex. Combining theory with simulations using data-driven in silico reconstructions will allow us to form quantitative predictions on how structure and network dynamics relate in health and disease.

Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recover the first and zero-th homology of the gluing from the respective homologies of the components.

A multifiltration is a functor indexed by $\mathbb{N}^r$ that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-modules that can occur as $R$-spans of multifiltrations of sets are the direct sums of monomial ideals., Comment: 21 pages, 3 figures

Machine learning models for repeated measurements are limited. Using topological data analysis (TDA), we present a classifier for repeated measurements which samples from the data space and builds a network graph based on the data topology. When applying this to two case studies, accuracy exceeds alternative models with additional benefits such as reporting data subsets with high purity along with feature values. For 300 examples of 3 tree species, the accuracy reached 80% after 30 datapoints, which was improved to 90% after increased sampling to 400 datapoints. Using data from 100 examples of each of 6 point processes, the classifier achieved 96.8% accuracy. In both datasets, the TDA classifier outperformed an alternative model. This algorithm and software can be beneficial for repeated measurement data common in biological sciences, as both an accurate classifier and a feature selection tool., Comment: 8 pages, 7 figures

We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector spaces leads to stabilization of discrete invariants. We develop theory behind this stabilization and stable rank invariant. We give evidence of the usefulness of this approach in concrete data analysis., Comment: 32 pages, 10 figures, appearing in SIAM Journal on Applied Algebra and Geometry 4-1

In this paper we study multidimensional persistence modules [5,13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction., Comment: Found Comput Math (2016)

The proofs of Ravenel’s conjectures [22] and their reinterpretations in the form of the classification of the homotopy idempotent functors of spectra commuting with telescopes by Devintaz–Hopkins–Smith [7, 12], were a culmination of a few decades of progress achieved in stable homotopy theory. The simplicity of this classification is remarkable. For each prime p, the category of restrictions of these functors to p-local finite spectra is isomorphic to the poset of natural numbers. The obtained invariant is called the Morava–Hopkins type. The stable classification was generalized to the classification of so-called Bousfield localizations of finite p-local spaces; see Bousfield [3]. This unstable classification is also remarkably simple. Bousfield showed that, in addition to the Morava–Hopkins type invariant, connectivity determines such functors.

We discuss constructions and categorical properties of various cellular approximation functors in the category of groups.

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages and, in particular, classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield–Kan homology completion tower \(z_k X\) whose terms are all X-cellular for any X.

Suppose that p is some chosen prime number. A topological space is said to be v∗torsion if its homotopy groups are concentrated at p and its vn-periodic homotopy groups vanish for n ≥ 1 (1.1). A similar definition can be made for spectra, and in this stable case it is easy to see that a spectrum is v∗-torsion if and only if it can be built (in the sense of [9]) from any finite spectrum with nontrivial mod p homology. The unstable analog of this statement is not true, but in this paper we provide a close substitute. For any space W , let BW denote the class of spaces which can be built from W , and T (BW) the smallest class of spaces which contains BW and is thick with respect to fibrations (1.6). We prove that a space X is v∗-torsion if and only if it belongs to T (BW) for any finite complex W with nontrivial reduced mod p homology. We go on to show that, although the category of v∗-torsion spaces is not closed under arbitrary homotopy colimits, it is closed in many cases with respect to realizations of simplicial objects. Finally, we consider the StoverBlanc simplicial resolution RW(X) of a space X, which comes up in a natural attempt to construct the W -cellular approximation CellW(X). We show that for appropriate W the fibres of the map from the geometric realization of RW(X) to CellW(X) are v∗-torsion. Since v∗-torsion spaces are in some respects negligible objects, this can be interpreted as saying that the realization of the Stover-Blanc construction is closer to CellW(X) than one might suspect.

Let ℳ be a simplicial model category and J : ℳ → ℳ a simplicial coaugmented functor. Given an object X, the assignment n↦J n+1 X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J s X = tots([n] ↦ J n+1 X). An object X is called J-injective if it is a retract of JX in Ho(ℳ) via the natural map. We show that certain homotopy limits of J-injective objects are J s -injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) → X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}s≥0→{totS X}s ≥ 0.

Suppose that A is a pointed CW-complex. The paper looks at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits, this is determined by an ordinal number called the complexity of B. Studying the complexity leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CWA(X) of an arbitrary space X.

Consider a push-out diagram of spaces C B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem., Comment: 22 pages, we took special care in this revised version in distinguishing fiber sets from single fibers, in indicating what we mean by the loop space on a possibly non-connected and unpointed space, thus smoothing the exposition

This thesis consists of two articles. Both articles concern homotopical algebra. In Paper I we study functors indexed by a small category into a model category whose value at each morphism is a weak equivalence. We show that the category of such functors can be understood as a certain mapping space. Specializing to topological spaces, this result is used to reprove a classical theorem that classifies fibrations with a fixed base and homotopy fiber. In Paper II we study augmented idempotent functors, i.e., co-localizations, operating on the category of groups. We relate these functors to cellular coverings of groups and show that a number of properties, such as finiteness, nilpotency etc., are preserved by such functors. Furthermore, we classify the values that such functors can take upon finite simple groups and give an explicit construction of such values.

We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial. We apply the technique of model approximations and use elementary properties of the category of spaces to be able to construct resolutions. We prove that the homotopy category of any monoidal model category is always a central algebra over the homotopy category of Spaces., Comment: Final version, almost as it will appear in "Algebraic and Geometric Topology"; 30 pages

We characterize the class of homotopy pull-back squares by means of elementary closure properties. The so called Puppe theorem which identifies the homotopy fiber of certain maps constructed as homotopy colimits is a straightforward consequence. Likewise we characterize the class of squares which are homotopy pull-backs "up to Bousfield localization". This yields a generalization of Puppe's theorem which allows to identify the homotopy type of the localized homotopy fiber. When the localization functor is homological localization this is one of the key ingredients in the group completion theorem., Comment: 18 pages

In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model approximation. Our key result says that if a category admits a model approximation then so does any diagram category with values in this category. From the homotopy theoretical point of view categories with model approximations have similar properties to those of model categories. They admit homotopy categories (localizations with respect to weak equivalences). They also can be used to construct derived functors by taking the analogs of fibrant and cofibrant replacements. A category with weak equivalences can have several useful model approximations. We take advantage of this possibility and in each situation choose one that suits our needs. In this way we prove all the fundamental properties of the homotopy colimit and limit: Fubini Theorem (the homotopy colimit -respectively limit- commutes with itself), Thomason's theorem about diagrams indexed by Grothendieck constructions, and cofinality statements. Since the model approximations we present here consist of certain functors "indexed by spaces", the key role in all our arguments is played by the geometric nature of the indexing categories., 95 pages with index

We study classes of modules over a commutative ring which allow to do homological algebra relative to such a class. We classify those classes consisting of injective modules by certain subsets of ideals. When the ring is Noetherian the subsets are precisely the generization closed subsets of the spectrum of the ring.

We describe the action of idempotent transformations on finite groups. We show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group. This is done in terms of the first two homology groups. We prove for example that except special linear groups, such an orbit can have at most 7 elements. We also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite.

This paper provides a comprehensive investigation of the cellular approximation functor cell A G , in the category of groups, approximating a group G by a group A . We also study related notions such as A -injection, A -generation and A -constructibility of a group G and we find several interesting connections with the Schur multiplier H 2 ( G , Z ) . Our constructions are direct and are given in a slow and detailed manner.

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower zkX whose terms we prove are all X–cellular for any X. As straightforward consequences, we show that if X is K–acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections PnX , and that any nilpotent space for which the space of pointed self-maps map .X; X/ is “canonically” discrete must be aspherical. Göran Gustafsson Stiftelse Fondo Europeo de Desarrollo Regional Ministerio de Economía y Competitividad